\(\int \frac {\cot (x)}{\sqrt {a+b \cot ^4(x)}} \, dx\) [62]

Optimal result

Integrand size = 15, antiderivative size = 41 \[ \int \frac {\cot (x)}{\sqrt {a+b \cot ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 \sqrt {a+b}} \]

[Out]

1/2*arctanh((a-b*cot(x)^2)/(a+b)^(1/2)/(a+b*cot(x)^4)^(1/2))/(a+b)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3751, 1262, 739, 212} \[ \int \frac {\cot (x)}{\sqrt {a+b \cot ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 \sqrt {a+b}} \]

[In]

Int[Cot[x]/Sqrt[a + b*Cot[x]^4],x]

[Out]

ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*Sqrt[a + b])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {a+b x^4}} \, dx,x,\cot (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {a-b \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right ) \\ & = \frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 \sqrt {a+b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\sqrt {a+b \cot ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 \sqrt {a+b}} \]

[In]

Integrate[Cot[x]/Sqrt[a + b*Cot[x]^4],x]

[Out]

ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*Sqrt[a + b])

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(\frac {\ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \sqrt {a +b}}\)	\(65\)
default	\(\frac {\ln \left (\frac {2 a +2 b -2 b \left (\cot \left (x \right )^{2}+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\cot \left (x \right )^{2}+1\right )^{2}-2 b \left (\cot \left (x \right )^{2}+1\right )+a +b}}{\cot \left (x \right )^{2}+1}\right )}{2 \sqrt {a +b}}\)	\(65\)

[In]

int(cot(x)/(a+b*cot(x)^4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/(a+b)^(1/2)*ln((2*a+2*b-2*b*(cot(x)^2+1)+2*(a+b)^(1/2)*(b*(cot(x)^2+1)^2-2*b*(cot(x)^2+1)+a+b)^(1/2))/(cot
(x)^2+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (35) = 70\).

Time = 0.39 (sec) , antiderivative size = 264, normalized size of antiderivative = 6.44 \[ \int \frac {\cot (x)}{\sqrt {a+b \cot ^4(x)}} \, dx=\left [\frac {\log \left (\frac {1}{2} \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + \frac {1}{2} \, a^{2} + \frac {1}{2} \, b^{2} + \frac {1}{2} \, {\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt {a + b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}} - {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )\right )}{4 \, \sqrt {a + b}}, -\frac {\sqrt {-a - b} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt {-a - b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )}\right )}{2 \, {\left (a + b\right )}}\right ] \]

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(1/2),x, algorithm="fricas")

[Out]

[1/4*log(1/2*(a^2 + 2*a*b + b^2)*cos(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a -
b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 -
b^2)*cos(2*x))/sqrt(a + b), -1/2*sqrt(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)
*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*co
s(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x)))/(a + b)]

Sympy [F]

\[ \int \frac {\cot (x)}{\sqrt {a+b \cot ^4(x)}} \, dx=\int \frac {\cot {\left (x \right )}}{\sqrt {a + b \cot ^{4}{\left (x \right )}}}\, dx \]

[In]

integrate(cot(x)/(a+b*cot(x)**4)**(1/2),x)

[Out]

Integral(cot(x)/sqrt(a + b*cot(x)**4), x)

Maxima [F]

\[ \int \frac {\cot (x)}{\sqrt {a+b \cot ^4(x)}} \, dx=\int { \frac {\cot \left (x\right )}{\sqrt {b \cot \left (x\right )^{4} + a}} \,d x } \]

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(cot(x)/sqrt(b*cot(x)^4 + a), x)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.41 \[ \int \frac {\cot (x)}{\sqrt {a+b \cot ^4(x)}} \, dx=-\frac {\log \left ({\left | -{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} {\left (a + b\right )} + \sqrt {a + b} b \right |}\right )}{2 \, \sqrt {a + b}} \]

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(1/2),x, algorithm="giac")

[Out]

-1/2*log(abs(-(sqrt(a + b)*sin(x)^2 - sqrt(a*sin(x)^4 + b*sin(x)^4 - 2*b*sin(x)^2 + b))*(a + b) + sqrt(a + b)*
b))/sqrt(a + b)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot (x)}{\sqrt {a+b \cot ^4(x)}} \, dx=\int \frac {\mathrm {cot}\left (x\right )}{\sqrt {b\,{\mathrm {cot}\left (x\right )}^4+a}} \,d x \]

[In]

int(cot(x)/(a + b*cot(x)^4)^(1/2),x)

[Out]

int(cot(x)/(a + b*cot(x)^4)^(1/2), x)